# Full text of "Analytical Mechanics"

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```220                      ANALYTICAL MECHANICS
182.   Experimental Definition of Moment of Inertia. — If in the experiment of the preceding section all frictional forces and torques are eliminated and then torques of different magnitudes are applied to the flywheel, it will be found that the torques are proportional to the angular accelerations produced; that is, if Gi, G2, etc., denote the torques obtained by multiplying the pull of the string by the radius of the axle and 71, 72, etc., the corresponding angular acceleration, then we shall find that the following relations hold :
Q = <" = *= ... =1,                      (II)
7l       72       73
where I is a constant which depends only upon the rotating system. In fact, as will be shown in § 186, it is nothing more or less than the moment of inertia of the rotating system. We have, therefore, the following definition for the moment of inertia of a body, in addition to the analytical definition given in Chapter VII :
The moment of inertia of a body about a given axis is a constant of the body, relative to the given axis, which equals the quotient of the torque applied by the angular acceleration obtained; both being referred to the given axis*
183.   Measure of Angular Kinetic Reaction. — It is evident from equation (II) that (?i, (?2, etc., which measure the angular kinetic reactions of the flywheel for the accelerations 71, 72, etc., are proportional to these accelerations.    Therefore the angular kinetic reaction of a body varies directly with the angular acceleration imparted.    If, on the other hand, a number of bodies of different moments of inertia are given the same angular acceleration, it is found that the kinetic reactions are proportional to the moments of inertia; that is,
Gl       (?2
* Note the striking similarity between this definition of moment of inertia. and the definition of mass given in § 94.```